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NOVEL TECHNIQUES FOR
PULSED FIELD GRADIENT NMR MEASUREMENTS
WILLIAM W. BREY
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
I would like to thank Janel LeBelle, Igor Friedman, and
Don Sanford for construction of the gradient coil
prototypes, and Jerry Dougherty for performing the
simulations of the coils they all helped to construct.
Stanislav Sagnovski, Eugene Sczezniak, Doug Wilken, and
Randy Duensing participated in many helpful discussions
concerning gradient coils. Debra Neill-Mareci provided the
excellent illustration of a gradient coil in Figure 22. For
their part in the microscopy project, thanks go to Barbara
Beck, Michael Cockman, and Dawei Zhou. Ed Wirth and Louis
Guillette provided the samples. For help measuring eddy
current fields I am grateful to Wenhua Xu, and to Steve Patt
for help with the software. Thanks go to my parents, Mary
Louise and Wallace Brey, and my brother, Paul Brey, for
encouragement and help with red tape. Paige Brey has my
special thanks for her extensive help preparing the thesis.
Katherine Scott, Richard Briggs, Jeff Fitzsimmons, and Neil
Sullivan enriched my graduate experience with their wide
knowledge and diverse interests. Thanks go to them for
their enthusiasm and for reading this thesis. Raymond
Andrew served as supervisory committee chairman. Thanks go
to Thomas Mareci for directing the research, for providing
financial and moral support, and for encouraging me to
pursue this work to its conclusion.
TABLE OF CONTENTS
ACKNOWLEDGMENTS .......................................... ii
ABSTRACT ................................................. v
GENERAL INTRODUCTION ....................................... 1
MEASUREMENT OF EDDY CURRENT FIELDS ....................... 5
Introduction ........................................ 5
Literature Review ................................... 11
Spin-Echo Techniques ................................ 23
Stimulated Echo Techniques .......................... 28
Results ............................................. 34
Conclusion .......................................... 41
GRADIENT COIL DESIGN ..................................... 46
Introduction and Theory ............................. 46
Literature Review ................................... 50
Field Linearity ..................................... 61
Efficiency .......................................... 62
Eddy Currents ....................................... 68
Coil Projects ....................................... 72
Amplifiers ....................................... 72
16 mm Coil for NMR Microscopy .................. 76
9 cm Coil for Small Animals .................... 79
15 cm Coil for Small Animals ................... 87
Concentric Return Path Coil .................... 98
SYSTEM DEVELOPMENT FOR NMR MICROSCOPY .................... 126
Introduction ........................................ 126
Literature Review ................................... 128
Instrument Development .............................. 133
Results ............................................. 149
Conclusion .......................................... 153
CONCLUSION ............................................... 155
REFERENCES ............................................... 156
BIOGRAPHICAL SKETCH ...................................... 162
Abstract of Dissertation Presented to the Graduate School of
the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
NOVEL TECHNIQUES FOR PULSED FIELD GRADIENT NMR MEASUREMENTS
William W. Brey
Chairman: E. Raymond Andrew
Major Department: Physics
Pulsed field gradient (PFG) techniques now find
application in multiple quantum filtering and diffusion
experiments as well as in magnetic resonance imaging and
spatially selective spectroscopy. Conventionally, the
gradient fields are produced by azimuthal and longitudinal
currents on the surfaces of one or two cylinders. Using a
series of planar units consisting of azimuthal and radial
current elements spaced along the longitudinal axis, we have
designed gradient coils having linear regions that extend
axially nearly to the ends of the coil and to more than 80%
of the inner radius. These designs locate the current
return paths on a concentric cylinder, so the coils are
called Concentric Return Path (CRP) coils. Coils having
extended linear regions can be made smaller for a given
sample size. Among the advantages that can accrue from
using smaller coils are improved gradient strength and
switching time, reduced eddy currents in the absence of
shielding, and improved use of bore space.
We used an approximation technique to predict the
remaining eddy currents and a time-domain model of coil
performance to simulate the electrical performance of the
CRP coil and several reduced volume coils of more
conventional design. One of the conventional coils was
designed based on the time-domain performance model.
A single-point acquisition technique was developed to
measure the remaining eddy currents of the reduced volume
coils. Adaptive sampling increases the dynamic range of the
measurement. Measuring only the center of the stimulated
echo removes chemical shift and B0 inhomogeneity effects.
The technique was also used to design an inverse filter to
remove the eddy current effects in a larger coil set.
We added pulsed field gradient and imaging capability
to a 7 T commercial spectrometer to perform neuroscience and
embryology research and used it in preliminary studies of
binary liquid mixtures separating near a critical point.
These techniques and coil designs will find application
in research areas ranging from functional imaging to NMR
As pulsed field gradient technology for NMR matures,
new and diverse applications develop. Pulsed Gradient Spin
Echo techniques allow the measurement not only of the bulk
diffusion tensor, but of the structure factor of the
sample.1 Editing techniques use pulsed field gradients to
simplify the complex spectra of biomolecules.2 Local
gradient coils allow functional imaging in the human head.3
NMR microscopy can require field gradients much larger and
switched more rapidly than conventional imaging
experiments.4 Localized spectroscopy allows chemical shift
information to be collected from specific voxels in a living
animal.5 This paper will address some approaches for
producing and evaluating pulsed field gradients.
A technique was developed to measure the eddy current
field that persists after a field gradient is switched off
and, based on the measurement, a filter to correct for the
eddy current field was designed. The technique, which
employs a series of experiments based on the stimulated
echo, was then used to evaluate the performance of the
1D. G. Cory and A. N. Garroway, Magn. Reson. Med. 14, 435, 1990.
2D. Brihwiler and G. Wagner, J. Magn. Reson. 69, 546, 1986.
3K. K. Kwong et al., Proc. Natl. Acad. Sci. 89, 5675, 1992.
4Z. H. Cho et al., Med. Phys. 15, 815, 1988.
5H. R. Brooker et al., Macn. Reson. Med. 5, 417, 1987.
filter. If an eddy current field persists during the period
when the NMR signal is detected, distortions in the spectrum
or image will result. The distortions are particularly
severe when chemical shift information is obtained in the
same experiment as spatial localization by encoding spatial
information in the phase of the NMR signal. It is important
to be able to measure the residual gradient field, which is
usually due to eddy currents in the metal structures of the
magnet, so that it can be corrected by changing the drive to
the gradient amplifier, or by whatever other technique is
available, and to evaluate the remaining uncorrected field
to estimate the distortion that will result in a desired
One way to avoid eddy currents for experiments such as
spatially selective spectroscopy is to employ actively
shielded gradient coils. Another, much simpler, approach is
to reduce the size of the gradient coil so that it is widely
separated from the eddy-current-producing structures in the
magnet. This approach is only possible when the clear bore
of the magnet is much larger than the volume of interest,
which is often the case. To make possible experiments, such
as spatially selective spectroscopy, that require rapidly
switched high intensity field gradients, I developed pulsed
field gradient systems based on reduced volume gradient
coils for a 2 T, 31 cm bore magnet used for small animal
studies. This magnet was replaced with a 4.7 T, 33 cm bore
magnet, and the gradient systems were adapted accordingly.
These pulsed field gradient systems offer much better
performance than the large and unshielded gradient system
supplied with the magnet, given their limitation on sample
size. I also developed a pulsed field gradient coil for a 7
T, 51 mm bore magnet used for NMR microscopy and
Another experiment which requires gradient coils to
perform exceptionally well is functional imaging of the
human brain. The head is much smaller than the whole-body
magnets in general use. A smaller coil can allow faster
switching to higher gradient fields, as well as reduce eddy
current fields. In order to get a gradient coil that is
matched to the size of the head, some provision must be made
to allow for the shoulders. Conventional designs, even
existing designs with a large linear volume, have current
return paths arrayed on both sides of the linear volume. A
coil matched to the size of the head would not fit over the
shoulders. A coil that trades radial linear region for
increased axial linear region is more appropriate. A design
utilizing concentric return paths was developed that
significantly improved the axial region of linearity. A
prototype was constructed and tested.
In order to perform NMR microscopy and pulsed field
gradient experiments, we adapted an NMR spectrometer and
probe for a 7 T, 51 mm bore magnet. The instrument included
a simple amplitude modulator to carry out slice selection.
A probe that allowed sample loading from above was
constructed. Artifacts were eliminated from the images. A
software interface that allows the user to set up an
experiment by entering values in a spreadsheet was
developed. Useful contrast was obtained on fixed biological
samples. Preliminary imaging experiments on both biological
and nonbiological systems were carried out.
MEASUREMENT OF EDDY CURRENT FIELDS
It is well known that, when a current pulse is passed
through a field gradient coil in a superconducting magnet,
eddy currents are produced in the conducting structures of
the magnet. Experiments such as diffusion-weighted imaging6
and multiple-quantum spectroscopy7 require that the eddy
current field be a much smaller fraction of the applied
field than do conventional spin-echo magnetic-resonance
imaging experiments. Strategies to reduce the eddy current
field consequently become increasingly important. The two
effective strategies are signal processing of the gradient
demand, known as preemphasis, and self-shielding of gradient
coils, which greatly reduces the interaction of the coil
with the metal structures of the magnet. Often, the two
techniques are used together. When the sample or subject is
substantially smaller than the magnet, another approach is
to minimize the size of the gradient coil. In order to
evaluate and improve the effectiveness of these three
strategies, it is desirable to have a technique to measure
eddy current fields. To implement the preemphasis, it is
necessary to measure the eddy current field in order to
6D. G. Cory and A. N. Garroway, Magn. Reson. Med. 14, 435, 1990.
7C. Boesch et al., Magn. Reson. Med. 20, 268, 1991.
cancel it. An eddy current measurement technique is also
useful in order to evaluate the possibility of performing a
given experiment with available hardware. In this chapter,
a technique for measuring and analyzing the time behavior of
eddy current fields is developed and experimental results
are presented. Some general physical considerations of eddy
currents are discussed, and existing techniques for eddy
current field measurement are reviewed.
An introduction to the Bloch equations will be
preliminary to a discussion of the effect of the eddy
current field on the nuclear magnetization. The Bloch
equations provide a phenomenological description of some
aspects of the behavior of spins in a magnetic field. Let M
be the bulk nuclear magnetization, y the gyromagnetic ratio,
By the polarizing magnetic field, and B1 the amplitude of
the radio frequency excitation field which has rotational
frequency o. T1 and T2 are the time constants associated
with longitudinal and transverse relaxation, respectively.
Mx = Y(BoMy + BlMz sin (ot) Mx/T2 
My = Y(BiMZ cos Ct BoMx) My/T2 
Mz = -y(BlMx sin Cot + BiMy cos Cot) (Mz Mo)/IT 
Instead of T2, the symbol T2* is used to denote the time
constant of apparent transverse relaxation when
inhomogeneity in BO is present. Neglecting the effects of
T1 and T2 and assuming B1 = 0, the equations can be
MX = TMyBo 
My = -'YMBO 
Mz = 0 
We can introduce a complex transverse magnetization M = Mx +
iMy so that
M = -iyMB0. 
Assume that Bo consists of a constant and a component
linearly dependent on position: B0 = BO+gx. BO is
independent of time and space, while g is quasi-static. If
we define m, the magnetization in the rotating frame, by
M = me-iBOt 
M = -iyBOM + me-iyBot 
Substituting back into Equation  gives
-iyBOM + ine-iyBot = -iyM(BO + gx). 
Simplifying Equation  yields
ne- iyBO t = -iyMgx. 
Combining Equation  with Equation  yields
m = -iygxm, 
which has the immediate solution
m(t) = m(to)e t 
If the magnetization has been prepared to a non-zero m(to)
by a radio frequency pulse, the evolution described by
Equation  is called a free induction decay (FID).
Consider the characteristics of a general eddy current
field. The eddy currents give rise to a magnetic field that
roughly tends to cancel the applied field of the gradient
coil. The spatial dependence of the eddy current field is
not exactly the same as the applied gradient field.8
The time behavior of the eddy current field is a
multiexponential decay, which can be seen by considering the
form of the solution to the differential equation governing
the decay of magnetic induction due to current flow in the
conductor. Maxwell's equations9 in a vacuum in SI units are
V B = 0 
V *E = 
V X E + -- = 0 
V X B oF00 DE = o0J 
where B is the magnetic induction and E is the electric
field, p is the charge density, e0 and go are the
permittivity and permeability of free space, and J is the
current density. We also assume Ohm's law, J = GE, where y
is the conductivity, assumed to be isotropic and
homogeneous. Taking the curl of both sides of Ampere's law,
Equation , neglecting the displacement current, and
using the identity
8R. Turner and R.M. Bowley, J. Phys E: Sci. Instrum. 19, 876, 1986.
9J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, New York,
V x (V x A) = V(V .A) V2A 
-V2B = g0V X J. 
Using Ohm's law to eliminate J for E, neglecting the
displacement current, yields
-V2B = 0CFV X E, 
so Equation  allows this to be expressed as
V2B = 0a 
The decay of the magnetic induction must be a solution to
this diffusion equation. Separation of variables gives
solutions for the time part having an exponential time
dependence. This makes it possible to correct for the
linear spatial term in the eddy current field with a linear
filter network. Such a network is known as a preemphasis
In a superconducting magnet, the conducting structures
involved are often at very low temperatures and hence have
much greater conductivity than might otherwise be expected.
For example, pure aluminum at 10 K has a resistivity of
1.93 x 10-12 J2-m, while at a room temperature of 293 K its
resistivity10 is 2.65 x 10-8 Q-m. The time scale of the
eddy current decay is directly proportional to its
conductivity, as can be inferred from Equation , so eddy
currents will persist 13,700 times longer in an aluminum
10D. R. Lide, (Ed.), CRC Handbook of Chemistry and Physics, 72nd
Edition, CRC Press, Boca Raton, 1991.
structure at 10 K than one at 293 K. In a commercial
aluminum alloy the conductivity will vary from that of the
pure metal, especially at low temperature, so the effect may
not be as great. In practice, the principal source of eddy
current fields is generally the innermost low temperature
aluminum cylinder, which is at approximately the boiling
point of liquid nitrogen, 77 K. The resistivity of aluminum
at 80 K is 2.45 x 10-9 Q-m, so the time constant is about
11 times greater than it would be at room temperature.
We consider the desirable characteristics for an eddy
current measurement technique. Our primary goal will be to
measure the eddy current field in order to evaluate the
feasibility of performing a given experiment, not to
compensate for the eddy current field. Therefore dynamic
range is more important than absolute accuracy. It must be
possible to measure eddy currents produced by specific pulse
sequences, probably by appending the eddy current
measurement experiment to the end of the sequence under
evaluation. It is also preferable to have a technique that
is insensitive to inhomogeneity so that no swimming is
necessary. Since the shim coil power supply may respond
dynamically to the gradient pulse and distort the measured
eddy current field, it is useful to be able to turn the shim
supply off. Experiments based on Selective Fourier
Transform11 and other chemical shift imaging techniques rely
11H. R. Brooker et al., Maan. Reson. Med. 5, 417, 1987.
for spatial localization on the integral of the eddy current
field, so it is desirable to have a measurement technique
that is based upon the integral of the eddy current field.
If possible, the technique should have no special hardware
Many workers have addressed the problem of eddy current
measurement and compensation in the literature. The two
aspects of the eddy current field to measure are the spatial
and time behaviors. We review publications that include
descriptions of eddy current measurements, although in most
cases the emphasis is placed upon the preemphasis
compensation process and its effectiveness, not the
measurement. The measurement process can be divided into
techniques that detect the derivative of the eddy current
field, those that detect the eddy current field itself, and
those that detect the integral of the eddy current field.
The derivative of the field is sensed by a pickup coil
consisting of turns of wire through which the changing flux
of the eddy current field produces an electromotive force
that is proportional to the rate of change of the field.12
A high impedance preamplifier boosts the signal. An analog
integrator is usually used to convert the measured voltage
into a quantity proportional to the field, although it is
possible to use digital integration. When used in a magnet
12D. J. Jensen et al., Med. Phys. 14, 859, 1987.
at field, the pickup coil is sensitive also to any change in
flux resulting from mechanical motion, which can contaminate
the measurement. Since the field of the main magnet is
generally about four orders of magnitude larger than the
eddy current field and the time scale of mechanical modes is
smaller than that of the longer time-constant eddy currents,
mechanical stability of the coil is crucial. Drift in the
analog electronics is another potential difficulty with the
pickup coil technique. Even with digital integration, the
preamplifier can experience thermal drift on time scales not
too different from the eddy current field. In spite of
these difficulties, pickup coils are simple to use and can
be used effectively to adjust preemphasis compensation.
They are used routinely to correct for eddy currents in
commercial, clinical MRI installations.13
A different approach to measuring the eddy current
field is through its effect on the NMR resonance. One
advantage here is that a pickup coil and its associated
hardware are not needed. These proportional techniques
measure a frequency shift in the NMR resonance that is
directly related to the eddy current field.14 From Equation
, the phase of freely-precessing magnetization in the
rotating frame at time t with respect to to can be written
13Personal communication, Dye Jensen.
14Ch. Boesch et al., Magn. Reson. Med. 20, 268, 1991.
= yxJ gdt'. 
The instantaneous frequency o(t), which can be defined as
the rate of change of the phase of 0 by (0(t) = d)/dt is
related to the eddy current field through the Larmor
equation (0 = yB. Magnetic field homogeneity is important
when using this approach, so that the FID will persist long
enough to obtain a meaningful measurement.
In another approach based on the NMR experiment, the
phase of the magnetization 0 is measured at a single point
in time. The phase at that point reflects the integral of
the eddy current field over certain intervals in the
experiment. Since only one point is sampled in each
experiment, many more experiments are required to map the
decay of the eddy current field than with the proportional
techniques. However, T2* and off-resonance effects do not
affect the usefulness of the technique. The experiment
proposed later is a single-point acquisition technique.
All the techniques surveyed were implemented for
unshielded gradient units, although preemphasis is typically
used on systems with shielded gradient sets as well.15
Boesch, Gruetter and Martin of the University Children's
Hospital in Zurich16,17 measure and correct eddy currents on
a 2.35 T, 40 cm Bruker magnet. The unshielded gradient set
has an inner diameter of 35 cm and a maximum gradient of 1
15R. Turner, Magn. Reson. Imaq. 11, 903, 1993.
16Ch. Boesch et al., Magn. Reson. Med. 20, 268, 1991.
17Ch. Boesch et al., SMRM 1989, 965.
G/cm. They use two NMR techniques to measure the eddy
current field. They interactively correct, using a 12 cm
diameter glass sphere filled with distilled water, and they
use no spatial discrimination in order to get all spatial
components. The experiment consists of a 2.5 s gradient
pulse of 0.6 G/cm followed by a train of 8 FIDs. There is a
20 ms delay between the time the gradient is switched off
and the first radiofrequency (RF) pulse. The RF pulses have
a 20 flip angle in order to reduce echo signals. The total
eight FID acquisition time is 200 ms. They solve the Bloch
equation for a sample with a single resonance frequency and
decay constant and extract
yABz(t) = (MydMx / dt MxdMy / dt) / (M2 + My) 
as an estimate of time-dependent Bo shift. They claim this
gives enough information for interactive preemphasis
adjustment. The one measurement they publish is of an
already corrected system and shows 7ABz(t) decaying from 2
to 0 ppm as time t increases from 20 to 200 ms. Glitches
are apparent at the ends of the FIDs.
To map the spatial variations, they place a stimulated
echo (STE) imaging experiment following the gradient pattern
of the experiment they want to analyze. The STE sequence is
applied with and without the preceding gradient pattern.
The difference in phase is considered to be due to the time
integral of the eddy current field in the interval between
the first two pulses of the STE sequence. A series of
slices tilted by multiples of 22.50 is obtained from the
same 12 cm diameter phantom. The images were phase
corrected. The phase of points along the z axis and on
circles around the z axis was measured and used as data for
a polynomial regression analysis to determine the
coefficients of the various spatial harmonics. A table of
the harmonic components following a 2.5 second x gradient
pulse of 0.3 G/cm is presented. The delay between the end
of the gradient pulse and the first RF pulse in the three
pulse STE experiment is 20 ms, and the delay between the
first and second RF pulse is 15 ms. The experiment was
conducted following adjustment of the preemphasis unit. In
decreasing order of magnitude, x, z, y, z2, xz2, xz, and x2-
y2 terms were present. The value of the B0 term was not
reported. Note that after x, the dominant terms should be
eliminated by the symmetry of the coil/cylinder system.
Only the x and xz terms would appear in an ideal system.
The presence of terms having even-order in x can be due to
two reasons. First, the terms may really exist due to
asymmetries in the magnet and gradient coil, crosstalk
between amplifiers, etc. Second, the spherical harmonic
analysis is highly sensitive to the point chosen to be the
origin, and the most favorable origin may not have been
A series of the phase-modulated images is presented as
well, with delays of 5, 20, 50, and 100 ms between the x
gradient and the STE imaging sequence. The images are all
from an already compensated system.
Van Vaals and Bergman of Philips Research Laboratories
in Eindhoven, the Netherlands,18,19 have a 6.3 T, 20 cm
horizontal bore Oxford magnet with 2 G/cm non shielded
gradients leaving a 13.5 cm clear bore. To measure the
eddy currents, they use a 4 cm diameter spherical phantom.
After swimming, they perform a simple "long gradient pulse,
delay 8, RF pulse, acquire" sequence. The gradient is
switched on for typically 3 s, but at least 5 times the
largest eddy current time constant. For various values of
8, the magnet is re-shimmed to maximize the signal during
the first 10 ms of the FID. The difference in shim values
with and without the gradient pulse is interpreted to be a
spherical harmonic expansion of the eddy current field.
Exact values of 8 are not listed, nor are tables of shim
values. Instead, the amplitudes and time constants of the
eddy current fields, as derived by a Laplace transform
technique, are given. Only the B0 and linear terms are
given; presumably only these terms were shimmed.
Jehenson, Westphal and Schuff of the Service
Hospitalier Frederic Joliot, Orsay, France, and Bruker,20
corrected eddy currents on a 3 T, 60 cm Bruker magnet. The
0.5 G/cm unshielded gradient coils had a clear bore of 50
18J. J. van Vaals and A. H. Bergman, J. Magn. Reson. 90, 52, 1990.
19J. J. van Vaals et al., SMRM 1989, 183.
20P. Jehenson et al., J. Macrn. Reson. 90, 264, 1990.
cm. They use the same type of multiple FID sequence as
Boesch, Gruetter and Martin, with an exponentially
increasing sampling interval and 30 sampling points. The
gradient prepulse is 10 s in length. The first FID is
sampled at 1.5 ms after switching off the gradient, and
sampling continues for 4 s using multiple FIDs. They plot
the measured field vs. the time with and without
compensation. They use a 1 mm by 3 mm water-filled
capillary positioned at +/- 5 cm to discriminate Bo and
linear terms. They do not consider crosstalk or higher-
order terms. They use the same Laplace transform technique
as van Vaals and Bergman, but they apply it iteratively to
get better correction.
Heinz Egloff at SISCO (Spectroscopy and Imaging
Systems, Sunnyvale, CA)21 used a pickup coil to measure eddy
current fields. To correct the B0 component of the eddy
current fields, he moved the gradient coils until the field
shift was eliminated.
Riddle, Wilcott, Gibbs and Price22 considered the
performance of a Siemens 1.5 T Magnetom. They measured the
instantaneous frequency do/dt of a 100 ml round flask
(presumably filled with water) following a 256 ms, 0.8 G/cm
gradient pulse. They present plots for imaging and
spectroscopy shims as well as for the gradient pulse. They
endorse do/dt as an indication of shim. It would seem to
21H. Egloff, SMRM 1989, 969.
22W. R. Riddle et al., SMRM 1991, 453.
work only for single-line samples, however. Following the
gradient pulse, the plot of d'/dt contains peaks that are
not explained. They may be an indication of the true do/dt,
or they may be artifacts from beginnings and ends of FIDs.
The sensitivity of the technique as presented here seems to
be about 1 Hz.
Hughes, Liu and Allen23 of the Departments of Physics
and Applied Sciences in Medicine at the University of
Alberta measured the eddy current fields of their 2.35 T, 40
cm bore Bruker magnet. After 57 delays ranging between 500
is and 2.5 s following a 0.2 G/cm gradient pulse the FID was
measured and the offset frequency of the line determined.
They placed a 13 mm diameter spherical water sample at +/-
1, 2, 4 cm along the axes of the radial gradients under
test. A four-exponential fit was applied to all six
locations simultaneously. The shortest time constant was
associated with the amplifier rise time. An interesting
plot shows that the field associated with each time constant
is essentially linear. The Bo fields associated with the
various time constants are different, however, suggesting a
unique isocenter for each time constant.
Zur, Stokar, and Morad24 of Elscint in Israel place a
doped water sample at +/- 5 cm from the center in the
direction of the gradient of the field. A train of 256 FIDs
is acquired after switching off the gradient. Each FID is
23D. G. Hughes et al., SMRM 1992, 362.
24Y. Zur et al., SMRM 1992, 363.
Fourier transformed, bandpass filtered, then inverse
transformed. The instantaneous magnetic field is obtained
from d)/dt. The digital filtering points to a problem with
phase measurements. The low-pass filters required to
eliminate Nyquist aliasing and to improve the signal-to-
noise ratio (SNR) distort the phase of the received signal.
Digital filtering enables one to recover the SNR ratio of a
small bandwidth without significant phase distortion.
Wysong and Lowe25 at Carnegie Mellon and the University
of Pittsburgh measured eddy current fields on a Magnex 2.35
T 31 cm magnet with unshielded gradient coils. A 1 cm
diameter sphere containing water doped to T-T2-~1 ms is
used. A 0.9 G/cm gradient is applied for 1.0 s, then ramped
down in 128 ps. A train of pulses of flip angle n/2 set 1
ms apart is applied for 1 second. One point is sampled for
each FID. With the system adjusted so the FID is in-phase
in the absence of a gradient field, the out-of-phase
component is proportional to sin(yABte-t/2) = yABte-t/2 for
small values of time and gradients.
Keen, Novak, Judson, Ellis, Vennart and Summers26 of
the Department of Physics, University of Exeter, propose
using a phantom slightly smaller than the imaging volume.
Having switched off the gradient, they delay a variable
time, then pulse and acquire the FID. The Fourier transform
25R. E. Wysong and I. J. Lowe, SMRM 1991, 712.
26M. Keen et al., SMRM 1992, 4029.
of the FID represents a projection of the phantom in the
quasi-steady eddy current field. Measuring the distance
between the peaks that appear as edge artifacts gives the
eddy current field.
Teodorescu, Badea, Herrick, and Huson27 at the Texas
Accelerator Center and Baylor College of Medicine measured
eddy current fields in their 4 T, 30 cm superferric self-
shielded magnet. The magnet was operated at 2.19 T. They
follow Riddle et al.28 in their measurement. A small
phantom is placed at various off-center locations. They use
a 0.8 G/cm gradient pulse of 15 ms and a 750 gs rise/fall
time. This is followed by an FID (or a series of them) that
is acquired for 20 ms. They compare this to the result
obtained from a pickup coil.
The eddy current field was measured with a sense coil
and analog integrator by Morich, Lampman, Dannels, and
Goldie.29 They used a Laplace transform approach to derive
correct parameter values for an analog inverse filter to
compensate for the eddy currents. The analog inverse filter
was of conventional design,30 placed at the input of the
gradient power supply. The theory was tested on an Oxford
Magnet Technology whole body superconducting magnet.
The approach is based on the ease with which a linear
system can be analyzed in the reciprocal space s defined by
27M. R. Teodorescu et al., SMRM 1992, 364.
28W. R. Riddle et al., SMRM 1991, 453.
29M. A. Morich et al., IEEE Trans. Med. Imac. 7, 247, 1988.
30D. J. Jensen et al., Med. Phys. 14, 859, 1987.
the Laplace transform. We can understand the calculation as
follows. Assume the gradient field for t>0 in response to a
unit step function is
g(t) = 1- ae-tw i i 
The amplitudes ai and time constants Ti can be determined
through a best-fit to experimental data. To determine the
inverse filter, the first step is to deconvolve the step
function to find the impulse response h(t), which can more
conveniently be accomplished by a multiplication in the
complex frequency space, s. The equivalent function G(s) is
obtained by a Laplace transform
1 N a
G(s) = _- N- ai 
S is + Wi
Then the impulse response in the s domain, H(s), is found
through the relation
G(s) = H(s)/s, 
H(s) = sG(s) = 1 N ais 
s + wi
is the impulse response. The inverse filter's impulse
response is just the reciprocal of the impulse response of
the eddy currents,
YH(s) = N 
s + Wi
The step response of the inverse filter, F(s), is the
convolution of a step function and the impulse response:
F(s) =- 
sH(s) N ais2
S + wi
The amplitudes bi and time constants vi of the inverse
filter can be read directly from the inverse Laplace
transform, f(t), of F(s):
f(t) = 1 + bie-t/vi 
The inverse Laplace transform was performed by matrix
inversion for a four-time-constant case using Gaussian
Now the appropriateness of these techniques to the
project of following the time evolution of the eddy current
field can be considered. Two of the techniques, those of
Egloff and Morich, involve the use of a pickup coil,
preamplifier, and integrator. We choose to confine
ourselves to NMR techniques. The procedures of Boesch, van
Vaals, Jehenson, Riddle, Keen, Hughes and Teodorescu require
swimming to correct for the inhomogeneity of Bo. The fact
that T2* must be reasonably long also limits the region
where eddy current fields can be measured to well inside the
active imaging volume. Wysong and Zur propose similar NMR
techniques that do not require swimming. In general,
however, it is samples with long relaxation times that are
most sensitive to small eddy current fields, and the use of
a sample with especially short (T~-T2-1 ms) relaxation times
is not an obvious way to detect low-level fields. The T2 of
the sample limits the duration of the interval in which
phase can be sampled. Another drawback is that the trains
of 7/2 pulses will produce stimulated echoes, even if T1 is
on the order of the interpulse separation. However, this
may be the most promising of the techniques surveyed.
Distortions in the phase of spectra spatially localized
with a two-pulse Selective Fourier Transform technique31
were observed by Mareci.32 He observed that the distortions
were reduced by lengthening the echo time, consistent with
the known behavior of field distortions due to eddy currents
induced in the metal structures of the magnet by the pulsed
gradient fields used for spatial localization. We consider
how a series of spin echo experiments identical except for
I I I
0 TE/2 TE
Figure 1. Two-pulse experiment with pulsed field gradient.
The long trailing edge of the gradient pulse indicates
distortion due to the eddy current field.
31H. R. Brooker et al., Magn. Reson. Med. 5, 417, 1987.
32T. H. Mareci, Personal communication.
increasing echo time (TE) gives an indication of the eddy
current field distortion as a function of time. Consider
the evolution of the rotating-frame magnetization m in the
presence of the gradient field g illustrated in the pulse
sequence in Figure 1. For O
Equation  so
-iyx f gdt'
m(t) = m(O)e 0 < t < TE / 2. 
We can also apply the result directly to describe the
magnetization's evolution following the n pulse. Let TE/2+
be the time just after the n pulse. Then
m(t) = m(TE / 2+)e TE / 2 5 t. 
The t pulse along x inverts the sign of the imaginary part
of m(t), equivalent to taking the complex conjugate:
iyx JTE/2 gdt'
m(TE / 2+) = m*(0)e 0 
Putting it together gives
iyxITE/2 gdt' -iE/2 gdt
m(t) = m (0)e e TE / 2 < t 
[ (TE/2 : ,
iyx gdt- gdt'
m(t) = m*(O)e J JTE/2 J TE / 2 < t. 
Measurement of the phase exactly at the center of the Hahn
echo should remove off-resonance effects, whether due to
chemical shift or field inhomogeneity. Now it remains to be
shown that measurements of the phase at a series of echo
times can be used to find g(t). If 00 is the phase without
a gradient pulse applied, then
(TE) 40 = yx E/2 gdt' J/ gdt] = yx2fTE/2 gdt' JTEgdt].
We define a function G(t) by
G(t) = yxJ gdt' 
which simplifies the expression above for ):
O(TE) 40 = yx[2G(TE / 2) G(TE)] 
By measuring 00 and measuring ) at a number of echo times,
we hope to be able to extrapolate the function G(TE), whose
rate of change gives the eddy current field. By performing
a series of experiments in which the values of TE are
related by successive powers of two (TEi1j = 2TEi), we can
obtain a series of coupled equations. Using the shorthand
0 (TEi) = 0j,
Oi+1 = y [2G +i] i = 1, 2, 
Inverting for G, yields
GI = [(4i+l 0o)/yx + Gi+1]/2 i = 1, 2, ... 
For large enough i, Gi = Gi+I, and the equation has an
immediate solution. The remaining Gi can be determined
recursively. The rate of change of G(TE) is the eddy
Experiments and subsequent data analysis have pointed
to several drawbacks in this approach. The first is that
the echo time TE limits the maximum length of the gradient
pulse. A gradient pulse long in comparison to the eddy
current decay time approximates a step function, which
simplifies the analysis of the eddy current response.33
However, lengthening the TE reduces the time resolution of
the experiment. Placing the gradient pulse before the
excitation pulse as in Figure 2 eliminates the problem and
decouples the length of the gradient pulse from the echo
Id2 0 d3 TE/2 TE/2
Figure 2. Gradtest v.1.2 is a spin-echo experiment for
measuring the eddy current field following a pulsed field
The above analysis assumes a point sample. Any real
sample has finite extent and will experience some dephasing,
and associated signal loss, as its phase evolves in the
gradient field. By not subjecting the transverse
magnetization to the gradient pulse but only to the eddy
current field, the dephasing effect is reduced. Another
drawback proved to be that the signal decayed due to T2
relaxation before Gi stabilized. With the gradient pulse
before the excitation, the condition Gi = Gi, could be met
for small values of TE. However, for large TE we could
33M. A. Morich et al., IEEE Trans. Med. Imaq. 7, 247, 1988.
assume that g = 0 while for small TE, g # 0. To solve for
the Gi it is necessary to know one of them in advance, so to
determine Gi for large TE, another experiment was performed.
TE was held fixed at a large value and d3, the interval
between the end of the gradient pulse and the RF excitation
pulse, was varied in steps of TE/2. A system of
simultaneous equations describes the phase obtained by
varying d3 in steps of TE/2, starting with d3 = 0:
O(TE + d3) 00 = yx[2G(TE / 2 + d3) G(TE + d3) G(d3)]
The problem of signal decay due to T2 is thus circumvented.
This technique could be used by itself or, as we used it,
only to obtain a starting point for varying TE.
A remaining difficulty is the ambiguity of phase
measurement. Phase can be directly measured only modulo
3600, but the accumulated phase in our experiment may be
much greater. One way around this difficulty is to reduce
the applied gradient so that we can be sure that our sample
rate is above the Nyquist limit, so that 4j+1 i < 1800
To get an upper bound that guarantees no phase ambiguity,
assume that the eddy current field has the same amplitude as
the applied field before the n pulse and zero amplitude
following the 7 pulse. Protons process at 4258 Hz/G. To get
a measurement for TE/2 = 512 ms without phase ambiguity
would, for a sample 1 cm from the center, require a gradient
pulse no greater than 0.000229 G/cm. Such a small gradient
pulse would result in no detectable phase accumulation in
practical cases. Experimental experience showed that it was
not simple to choose in advance a gradient amplitude that
would result in measurable phase accumulation, but no phase
ambiguity, at all echo times. Instead, we repeated the
experiment for a series of increasing gradient amplitudes.
For phase changes of less than 3600, the phase doubles as
the gradient doubles. We could keep track of phase
accumulations greater than 3600, thereby decreasing the
minimum detectable eddy current field.
Stimulated Echo Techniques
The stimulated echo (STE) has advantages over the spin
echo as the basis of an eddy current field measurement
experiment. Consider the stimulated echo sequence Gradtste
in Figure 3. The "e" at the end of the pulse sequence name
indicates that this is a stimulated echo experiment. A third
pulse is required to excite a stimulated echo. The
magnetization of interest is flipped into the transverse
plane by the first RF pulse, where it accumulates phase
RF n n
I I I I I
t T t
grad decay 1
Figure 3. Diagram for Gradtste, a three pulse stimulated
echo experiment for measurement of the eddy current field.
shift due to static field inhomogeneity and eddy current
fields. Then, stored by the second RF pulse along the z
axis, the magnetization accumulates no more phase until the
final RF pulse tips it back into the transverse plane. The
phase accumulation due to static inhomogeneity now unwraps,
resulting in the stimulated echo. If tj is long enough,
there is essentially zero eddy current field in the second
T, so the phase accumulated due to eddy current fields in
the first T is preserved.
It is possible to follow the eddy current decay by
incrementing either tdecay or T between experiments. If T is
incremented, the procedure for determining the eddy current
field is similar to that for spin echo experiments. The
phase shift for two experiments with different T is
subtracted to get the integral of the eddy current field in
the time between the earlier and later T. A more direct
approach is to increment tdecay between experiments, keeping
T small. Using this approach, each experiment yields the
integral of the eddy current field over a short interval T.
Dividing by T yields the average eddy current field in the
Two advantages of the STE are immediately evident. A
single STE experiment can be directly related to phase
accumulation in a single interval, eliminating the need for
the recursive data analysis or simultaneous equations
associated with the spin echo technique. This would also
seem to make the choice of gradient pulse amplitude more
straightforward. Since tI is limited by T1, which is
generally longer than T2, it is possible to sample with
smaller residual gradient field than in the spin echo
The eddy current field is subject to a multiexponential
decay. The integral of a multiexponential decay is another
multiexponential decay. We can expect these functions to be
reasonably smooth. That is, if we notice that the phase is
not changing much between delay increments, we could either
increase the delay increment or increase the amplitude of
the gradient pulse. This is a form of adaptive sampling,
since the sampling strategy for the gradient field depends
upon its behavior. The sampling technique should be capable
of following the residual field decay when preemphasis is
used, and in this situation the field will not in general
decay monotonically, since some of the decay components may
be overcompensated. Therefore the adaptive sampling must
also be able to decrease sensitivity when needed.
Since the eddy current field generally changes most
rapidly at short times, varying T to keep the measured phase
shift approximately constant for each value of tdecay yields
less densely spaced measurements when the field is changing
slowly. We have implemented such an adaptive sampling
technique by writing a recursive macro Adgrad in the Varian
MAGICAL language to perform a series of measurements in
which T is varied to "lock" the phase shift to 45. The
macro functions as a command to the Varian program "VNMR"
through which the spectrometer is controlled. Adgrad allows
the automatic measurement of the eddy current field over a
large dynamic range. Forty-five degrees is large enough to
measure with enough precision and yet small enough to
minimize the possibility of aliasing. The values of phase
" Adgrad ( t max) "
Figure 4. Flow chart of the macro Adgrad, which executes
adaptive sampling of the eddy current field. The dotted
portion is not part of the macro.
shift A0 and T are easily reduced to a plot of the eddy
current field vs. time. A flow chart of Adgrad is found in
Figure 4. It is most easily explained in the context of the
whole experimental procedure. The user notes the phase of
the STE for an experiment with gph, the value of the
gradient pulse, set to zero. He then selects a combination
of T, tdecay, and gph that results in phase accumulation of
about 450 and acquires an FID. He also removes the file
"phase.out" if it remains from a previous session. Then he
executes the macro Adgrad(tmax, o0), where tmax is the value
of tdecay at which the macro will stop and 00 is the phase
with gph = 0.
Adgrad first calls the macro Calcphase to compute the
phase A0 at the center of the acquisition window (that is
also the center of the FID) for the data already in memory.
Adgrad then stores the values of A0 and T as the first line
in the output file "phase.out." Next, Adgrad tests to find
if tdecay > tmax. If so, it ends the experiment. This
should not occur on the first pass through the test. In the
following two steps, Adgrad sets up the timing for the next
experiment. The new tdecay is set to be greater than the
old by T, to provide for a contiguous series of intervals t.
The new T is set so that if the eddy current field remains
constant, the next measurement will yield a phase A0 of 45.
Now the measurement is started. Following the measurement,
the macro calls itself and the process repeats. When the
tdecay > tmax test is passed, Adgrad returns control to the
Two other adaptive sampling macros have been developed
for eddy current testing. Adgrad2 changes both T and gph to
lock the phase to 45. The resulting series of experiments
are more closely spaced in time than Adgrad. Using Adgrad2,
it is possible to follow the eddy current field over a wider
range of values than with Adgrad. However, linearity error
in the digital-to-analog converter or nonlinear amplifier
response will be reflected in error in the eddy current
field. Adgradl80 locks the phase to 1800. It can only work
when the phase accumulation is monotonically decreasing yet
never changes sign, which is true for the uncompensated
gradients. Otherwise, Adgradl80 may lose its lock. If the
error in the measured angle is constant, the accuracy of the
technique, when applicable, should be about 5 times better
than for Adgrad or Adgrad2.
In the preliminary data analysis, we assumed that the
eddy current field was essentially constant over the
sampling interval T, so that g = AO/yxT. An Excel
spreadsheet was used to reduce and analyze the data and plot
the results. An example is shown in Figure 5. It is a plot
of the average field in each of the measurement intervals.
Since the field is dropping exponentially, not linearly,
fits to the mean value will have systematic errors. A
better way is to assume a multiexponential decay of the eddy
current field, and then calculate what phase will be
measured in the STE experiment. If we define OI(t) as the
total phase shift from t = 0 to tn for gph = 1, then
(D(t) = 
This phase shift is just, for a single experiment,
*((t) = yx g(t'dt' 
Now we assume that the eddy current field can be described
by a three-time-constant decay,
g(t) = Ae- tta + Be-t/tb + Ce- c. 
Integration gives a function to which the measured phase can
D (t) = yx[taA(l e-tta) + tbB(l e-t/tb) + tc( etc)].
Eddy current measurements were made on several gradient
coils of practical interest. Tests of the Oxford gradient
coils in the 2 and 4.7 T magnets were conducted. For the
4.7 T magnet, the eddy current measurements were used to
adjust the preemphasis network. Measurements of the eddy
current fields associated with home-built gradient coils
were also made. The detailed design and construction of the
coils, on 9 and 15 cm former, is described in the following
Initial measurements were made using the spin-echo
technique of Gradtest vl.2. A 5 mm NMR tube with about 5 mm
of H20 trapped by a vortex plug was used as a sample and
placed 1.7 cm from the center of a 2 T, 31 cm horizontal-
bore magnet (as measured from an image). The Oxford Z
gradient in the Oxford 2 T magnet was pulsed to a value of
1000 units or 1 G/cm. The manufacturer-installed
preemphasis filter was in place. A d3 array with four
elements was used to establish the phase value for large
echo times via matrix inversion of simultaneous equations.
An echo time array resulting in a series of coupled
equations was used to work back to 1 ms. The resulting plot
is shown in Figure 5. The bumpiness of the plot may be due
to the preemphasis. Data points are plotted in the center
of the interval for which they represent the average
0 --- I I-Illi'- -
0 100 200 300 400 500 600 700 800 900
Figure 5. Eddy current field as a fraction of applied field
for Oxford gradient coil.
Stimulated echo measurements using the pulse sequence
Gradtste and the macro Adgrad were conducted for the Oxford
gradients as well as for the 9 cm home-built gradient coil
in the 2 T, 31 cm diameter magnet. The eddy currents for
the Oxford gradients were measured with the manufacturer-
installed preemphasis filter in place. The 9 cm coils had
no preemphasis. A 5 mm NMR tube with about 5 mm of H20
trapped by a vortex plug was used as a sample and placed
between 1 and 2 cm from the center of the magnet. The
center of the sample was determined from an image. In all
cases, tgrad = 2 s, dl = 2 s, tj = 0.5 s, and two averages
were acquired. The parameter T was set to 4 ms and tdecay was
1 ms for the initial experiment. The data were analyzed in
Excel spreadsheets. In the plots of field vs. time given in
Figures 6 and 7 for the Oxford and 9 cm coils respectively,
the average of the eddy current field over the sampling
interval is plotted against the middle of each sampling
interval. The eddy current field is represented as a
percentage of the applied gradient. Note that without
preemphasis, the eddy current field due to the 9 cm coil
declines monotonically, while the preemphasis filters
contribute to the measured field of the Oxford coils. For
the 15 cm coil tested in the 2.0 T magnet, eddy current
measurement was used to calculate values for an inverse
filter. The coils and samples were removed between the
experiments before and after preemphasis. The Techron 7540
amplifiers were used to drive the coils in current mode.
Hm r in
_ --,- I
I~ 11 I -
a) 00 r- w
in m (N H-
S Al 0Co
-mm "m **.
r- 0 0 0 0
(%) ( )B
, -- I .
The same sample and RF coil were used as in the Oxford and 9
cm tests. In all cases, tgrad = 0.5 s, tL = 0.5 s, the
sample position was between 1 and 2 cm from the center, and
the position was measured in an image. The data were
0.1 0.2 0.3 0.4
0.3 0.4 0.5
= t (s)
0.1 0.2 0.3 0.4
Figure 8. Eddy current field of 15 cm gradient coil set in
2.0 T magnet system before (upper curve) and after
compensation (lower curve). a) X coil; b) Y coil; c) Z coil.
acquired with Adgrad2, that changes gph as well as T to keep
the phase locked. Eight averages were acquired. The eddy
current field was measured out to 1 s, although the plots in
Figure 8 only show 0.5 s. The data were analyzed both with
the average-field technique used in the Excel (Microsoft,
Inc.) spreadsheet and with a multiexponential curve fit in
Mathematica (Wolfram Research, Inc.). The curve fits seemed
more satisfactory, and are shown in Figure 8. The lower
curves represent the eddy current field after compensation.
The curves plotted are the derivatives of the exponential
curves that were fitted to the raw data. The preemphasis
filter amplitudes and time constants were taken to be those
of the eddy current field itself. This procedure should
0(t) (degrees) Echo Phase Shift
0.2 0.4 0.6 0.8 1
Figure 9. Fit to raw data of eddy current field of Oxford Z
gradient field for 4.7 T magnet system before compensation.
tend to underestimate the preemphasis required, but since
the unshielded eddy current fields were already less than 5%
of the applied field, the error is not severe.
The 4.7 T magnet that replaced the 2 T 31 cm magnet did
not have manufacturer-installed preemphasis, so the eddy
current measurement techniques were applied to design an
appropriate preemphasis filter. Since the uncompensated
eddy currents were on the order of 50% of the applied field,
the approximation used to compensate the 15 cm coil would
not be effective. An inverse Laplace transform technique
was used to design the filters. The technique was
implemented through the symbolic inverse Laplace transform
capability of Mathematica. An example of a multiexponential
fit to the raw phase accumulation performed with Mathematica
is shown in Figure 9. Eddy current fields before and after
compensation are presented in Figure 10. The upper curves
represent the field before, and the lower curves after,
preemphasis. For the Y coil, the procedure was repeated a
second time to obtain an additional reduction of the eddy
current field. The lowest curve in Figure 10 (b) represents
the eddy current field after the second pass of eddy current
A technique to measure the eddy current field of a
pulsed field gradient based on the phase of the stimulated-
echo NMR signal has been proposed. Experimental
0 t (s)
a) 0.2 0.4 0.6 0.8 1
0 t (s)
b) 0.2 0.4 0.6 0.8 1
0 t (s)
C) 0.2 0.4 0.6 0.8 1
Figure 10. Eddy current field of Oxford gradient coil in
4.7 T magnet system before (upper curve) and after
compensation (lower curve). a) X coil; b) Y coil. Lowest
curve was acquired after second-pass preemphasis; c) Z coil.
verification consists of measurements of the eddy current
field before and after preemphasis. The level of the eddy
current field after preemphasis can be interpreted as an
upper limit on the error bar of the measurement. It is only
an upper limit, since other errors also contribute to the
residual eddy current field. Any error in the values of
timing components in the preemphasis filter will add to the
residual eddy current field. Also, any distortion in the
amplifier will reduce the effectiveness of the compensation,
since the filter was designed based on the assumption that
the amplifier is linear. Error in the eddy current
measurement technique itself might be due to other echo
terms than the stimulated echo contributing to the signal.
However, experiments have shown that other echoes are
essentially negligible due to a combination of favorable
timing and phase cycling. In the case of Adgrad2, which
scales gph as well as T, it is clear that some error is due
to inaccuracies in the digital-to-analog converter (DAC)
output level. The applied gradient is then not proportional
to the DAC code, and so there is an error in normalizing to
the applied gradient. Error in the curve fits may be
significant, since in a multiple-exponential fit it is
difficult to get an accurate fit if the time constants are
not widely separated. Note that second-pass adjustment of
the preemphasis was more effective in reducing the residual
eddy current field.
The technique came out of a need to quantify phase
distortions in localized spectroscopy. It is therefore
better-suited to measuring the time integral of the eddy
current field than the field itself, and it is the integral
of the field that gives rise to errors in phase-sensitive
techniques such as SFT. It is often useful to employ the
basic stimulated echo experiment without adaptive sampling
to quantify the integral of the eddy current field over an
interval, and allow one to predict the resulting phase
The adaptive sampling algorithm is able to follow eddy
current fields that are not simply monotonically decreasing.
I found experimentally that if the angle became much
different than 450, the values for T would bounce around a
lot before stabilizing. This is probably due to the control
being purely proportional. Introducing an integral term
The relatively large eddy current field produced by the
15 cm Z gradient compared to the X and Y channels is due to
its extended-linearity design, which locates the currents
farther from the region of interest than a Maxwell pair.
The relatively large eddy current field produced by the 9 cm
Y gradient compared to the X and Y channels may be due to a
problem with centering the gradient coils in the bore. The
measured field gradient would then depend strongly on the
position of the sample.34
The contrast in eddy current field between the large
and small coils is clear. There is a factor of about 10 in
eddy current field between the 15 cm coil and the larger
Oxford coil. There is a factor of about 180 in eddy current
field between the 9 cm coil and the Oxford gradient coil set
34D. J. Jensen et al., Med. Phys. 14, 859, 1987.
in the X and Z channels. Experimental evidence demonstrates
the advantage in eddy current field obtainable with reduced
size gradient coils.
GRADIENT COIL DESIGN
Introduction and Theory
Although virtually all NMR measurements rely on
auxiliary field coils, there has been comparatively little
published work on the design and analysis of shim and field
gradient coils compared to that for radio frequency coils.
However, high levels of performance have become increasingly
important for these low-frequency room-temperature coils on
several frontiers of the NMR technique. Three of these
areas are gradient coils for NMR microscopy, coils for
spatial localization of spectra, and local gradient coils
for functional imaging of the human brain.
The simple forms of discrete element coil designs have
linear regions that are about 1/3 of the coil radius.35
Therefore the gradient coil must be considerably larger than
the sample. Increasing the linear region would allow
smaller coils to be used, generally improving efficiency and
decreasing eddy current fields. Several approaches are
available to increase the region of linearity. Adding
discrete elements to cancel more high-order terms in the
harmonic expansion has been done successfully by Suits and
Wilken.36 Continuous current density coils have also been
35F. Romeo and D. I. Hoult, Magn. Reson. Med. 1, 44, 1984.
36B. H. Suits and D. E. Wilken, J. Phys. E: Sci. Instrum. 23, 565, 1989.
designed with linear regions that are a large fraction of
the radius.37 We have tried to take a fresh approach,
combining aspects of both continuous and discrete designs.
For a solenoidal main magnet, available radial gradient coil
designs are longer and less efficient than axial designs, so
we have chosen to concentrate on the radial case.
/ 0 y
Figure 11. The coordinate system used in the text.
An appropriate starting point to find a new radial
gradient coil design might be: what current distribution on
the surface of an infinitely long cylinder would produce a
field in which the axial component is linearly proportional
to the radial position, Bzo x? To describe surface
currents and fields, we introduce the three coordinate
systems described by Figure 11. Any point can be described
in any of three orthogonal coordinate systems. In the
Cartesian system a point is described by its location along
the three axes (x, y, z). In the spherical system, it is
described by two angles and the distance from the origin:
37R. Turner, J. Phvs. D: ADpp. Phys. 19, L147, 1986.
(r, 0, )) In the cylindrical system, the point is
described by (p, ), z). It can be easily shown that an
azimuthal component of the surface current, J0, proportional
to cos) and independent of z produces the desired spatial
dependence. Neglecting for the moment the problem of
current continuity, there are two possible approaches to
achieving the cos) angular dependence. First, it can be
approximated by superimposing azimuthal currents with no
axial component. The solutions are exactly the same as for
discrete filamentary currents. The first approximation, the
1200 arc familiar from the so-called Golay double-saddle
design,38,39 is shown in Figure 12(a). This class of
designs has been called the "Golay Cage" because of its
correspondence to the double-saddle design. Higher-order
approximations utilizing superimposed arcs are derived by
Suits and Wilken.40 The other approach is to use our
freedom to choose any axial current to meet J4o cos# by
varying the current direction, for example, J4Oc cos#), Jzo
sin). This approach leads to the Cosine Coil shown in
Figure 12(b). Note that in Figure 12 the return paths are
located away from the active volume of the coil. For a coil
of practical length, the current return paths can
significantly reduce and distort the gradient field.
38F. Romeo and D. I. Hoult, Magn. Reson. Med. 1, 44, 1984.
39M. J. E. Golay, Rev. Sci. Inst. 29, 313, 1958.
40B. H. Suits and D. E. Wilken, J. Phys. E: Sci. Instrum. 23, 565, 1989.
Figure 12. Two radial gradient coils, a) The Golay Cage
Coil; b) Cosine Coil.
Another approach to current return paths is possible if
we relax the requirement that the current is confined to
the surfaces of cylinders. The current return paths can be
located in the same plane as the azimuthal current paths. A
gradient coil can be constructed of a stack of planes
approximating a current sheet, such as shown in Figure 34
(a) on page 111. The planes include radial as well as
azimuthal current elements. The radial currents do
contribute to the axial magnetic field. It happens that the
third-order harmonic terms eliminated by using 1200 arcs are
independently zero for the radial currents connecting the
arcs. These Concentric Return Path (CRP) Coils can have a
linear region that can be increased in length by stacking
more planes together. The overall combined coil structure
can also be very short, since the return paths do not
require extra length. In order to improve the linear region
beyond that produced by a constant current density along z,
we can adjust the relative current or position of each
This literature review will be focused on efforts to
increase the useful volume of a gradient coil, to optimize
its performance, and to understand the eddy current field
associated with a switched gradient it produces. The
specific requirements of coils of interest for functional
imaging of the human head are discussed, along with several
approaches to meeting those requirements.
Gradient coils can be grouped into two broad
categories: those made up of discrete current elements as in
Figure 13, and those approximating a continuous current
density. The former include the original NMR shim coil
designs,41'42 while the latter approach has been used to
make possible actively shielded gradient coils.43
Anderson described a set of electrical current shims
for an NMR system based on an electromagnet.44 The coils
were located in two parallel planes, one against each
poleface, to allow access to the sample. Each coil was
designed to produce principally one term in the spherical
harmonic expansion of the field. The orthogonality of the
41W. A. Anderson, Rev. Sci. Inst. 32, 241, 1961.
42M. J. E. Golay, Rev. Sci. Inst. 29, 313, 1958.
43P. Mansfield and B. Chapman, J. Maan. Reson. 66, 573, 1986.
44W. A. Anderson, Rev. Sci. Inst. 32: 241 1961.
expansion ensured relatively independent adjustment of the
current in the various coils.
Techniques for designing higher-order shim coils for
solenoidal magnets were set forth by Romeo and Hoult.45
Coils are designed by expanding the Biot-Savart integral for
Bz, the axial component of the field, in a spherical
harmonic series about the center of the coil for simple
filamentary building block currents.
Bz(r,9,()) = XAl,mPi(cos )ei. 
The functions Pmf(cos ) are the associated Legendre
functions. As building blocks are added in the form of arcs
on the surface of a cylinder, more terms in a spherical
harmonic expansion of the field can be set to zero. The
designer connects the building blocks in such a way as to
satisfy the requirement of current continuity, which is not
built into the Biot-Savart law. By setting each undesired
term in the harmonic series to zero, a system of equations
results. The solutions are the current, length, and
position parameters of the coil designs. A Maxwell pair, as
shown in Figure 13(b), is composed of a loop placed at 0 =
600 and another having opposite current direction placed at
8 = 120. This separation is required to cancel the (1, m)
= (3, 0) term, while the odd symmetry cancels the (2, m) and
(1, m # 0) terms. The desired (1,0) term remains. The
45F. Romeo and D. I. Hoult, Maan. Reson. Med. 1, 44, 1984.
simplest coil producing a gradient perpendicular to the axis
of the cylinder is the double saddle or "Golay" coil
illustrated in Figure 13(a). The arcs all subtend 1200 and
are placed at the four angles 01 = 68.70, 02 = 21.30, 1800-1,
and 1800-2, where they produce an (1, m) = (1, 0) term but
no (1, m) = (3, 0) term. The relative current directions
are shown in Figure 13(a). A family of solutions exists for
which the sum of the (1, m) = (3, 0) terms produced by the
arcs cancels, but the (1, m) = (3, 0) terms produced by each
arc are not necessarily zero. We designate such coil
designs by the two angles 01 and 02, so that the design
above would be described as 68.70/21.30. Adding additional
current elements to the coils adds degrees of freedom to
the system of simultaneous equations, and makes it possible
to cancel more terms.
Adding another pair of loops adds two more degrees of
freedom (the current and position of the new loops), and
makes it possible to cancel higher-order terms including (5,
0). Note that the equations are not linear, so for a large
number of current elements the procedure becomes unwieldy.
For shim coils, it is less important to improve the linear
region of a first-order or gradient coil than to design
additional coils whose lowest-order terms are of
increasingly high order. The simple saddle coil in Figure
13(a) designed by this technique has a useful volume with a
radius of about 1/3 that of the cylinder.
Figure 13. Field gradient coils that use discrete
filamentary current elements. a) Double-saddle 68.70/21.30
coil to produce the radial field gradient x or y; b)
Maxwell pair produces the axial, or z, field gradient.
The approach is best suited to cases where the gradient
coil is much larger than the sample, since the harmonic
series approximation to the field converges more rapidly
near the center of the coil. Although in theory the current
elements are lines, in practice they do have finite
dimensions, especially where large field intensity is
required. Including the wire diameter would greatly
complicate the design procedure. It is natural to use this
technique to design the shim coils mentioned above, where it
is conventional to have separate adjustments for as many as
twelve or more terms in the harmonic series. The coils
designed this way have the advantage of simplicity of
The building block approach was successfully extended
by Suits and Wilken46 to use discrete wires to produce a
constant field gradient over an extended region. They
evaluated designs for cylinders with the polarizing field
both parallel and perpendicular to the axis. To improve the
useful volume of the radial gradient coil, they superimposed
four saddle coils. The available degrees of freedom then
included the number of turns in each of the four coils, the
angular width of the four arcs, and the axial positions of
the arcs. Systems of nonlinear equations result that were
solved to null desired terms in an expansion of the field in
orthogonal functions. Numerical plots demonstrate that the
useful volume was extended to about eight times that of the
simple saddle coil. In each case, the volume was
nonspherical. The problems of extending this approach
further are that larger systems of nonlinear equations are
increasingly difficult to solve, and that the orthogonal
expansions do not converge rapidly away from the center of
Bangert and Mansfield47 designed a gradient coil in
which the wires were included in two intersecting planes.
The wires in each plane formed two trapezoids symmetrically
placed about the z axis. By setting the angle between the
planes to 450, the third-order terms in the magnetic field
46B. H. Suits and D. E. Wilken, J. Phys. E: Sci. Instrum. 23, 565, 1989.
47V. Bangert and P. Mansfield, J. Phys. E: Sci. Instrum. 15, 235, 1982.
Approaches based on a continuous current density are
most often used for gradient coils, where performance and
linear volume are more important than simplicity. The
impetus for these coils was the echo-planar imaging
technique of Mansfield,48 which requires high intensity
field gradients switched about an order of magnitude faster
than conventional Fourier imaging. Also, shielded coils are
useful in other imaging experiments that require rapidly
switched gradient fields, and in volume localized
spectroscopy. Without shielding to cancel the external
field, the higher frequency and intensity lead to greater
eddy currents in the cryostat and magnet, that in turn
distort the linearity and time response of the field. Using
current on two concentric cylinders, it is possible to
produce a linear field inside the inner cylinder and zero
field outside the outer cylinder. Continuous current
density coils can be designed to have a large linear region,
and, since current flows on the surface of the whole
cylinder, high efficiency.
A Fourier transform technique was applied by Turner to
design gradient coils that approximate a continuous current
distribution. The approach arose from consideration of the
eddy currents induced on cylindrical shields concentric to
gradient coils made up of discrete arcs.49 Expansion of the
Green's function in cylindrical coordinates was a natural
48p. Mansfield and I. L. Pykett, J. Macn. Reson. 29, 355, 1978.
49R. Turner and R. M. Bowley, J. Phys. E: Sci. Instrum. 19, 876, 1986.
approach to calculating the eddy current distribution. It
was then possible to write the field produced by a general
current distribution on the surface of a cylinder as a
Fourier-Bessel series.50 An inverse Fourier transform of
the Fourier-Bessel series allowed the current to be
expressed in terms of the desired field on the surface of an
imaginary cylinder. The field must satisfy Laplace's
equation to allow the existence of the inverse Fourier
transform. So by specifying the inverse Fourier transform
of the desired field, the current distribution required to
generate that field could be calculated. The continuous
distribution of current is approximated by discrete wires.
The wires are placed along the contour lines of integrated
current. Although the principal application of the
technique was shielded gradients, unshielded coils having
extended linearity were also designed. For example, a
radial gradient coil is reported to have a gradient uniform
to within 5% over 80% of the radius and a length of twice
the radius. The overall length of the coil is about 9 times
It was pointed out by Engelsberg et al. for the case of
a uniform solenoid that the homogeneity of the coil depends
strongly on the radius of the target cylinder.51 They note
that the field has the target value only on the surface of
the target cylinder. For example, in order to achieve a
50R. Turner, J. Phvs. D: Appl. Phys. 19, L147, 1986.
51M. Engelsberg et al., J. Phys. D. 21, 1062, 1988.
homogeneous field along the axis of the solenoid, the target
cylinder should be as narrow as possible. The effect is
especially pronounced at the ends of the target cylinder.
The importance of functional imaging of the human brain
and its reliance on the echo planar imaging technique puts
special demands on the rise time and field of the gradient
coil. The fact that smaller coils will be more efficient
and less affected by eddy currents has motivated several
workers to design gradient coils that will fit closely over
the head. To use a small gradient coil it is necessary to
have extended linearity in the radial and axial directions.
For a head coil, extended axial linearity is especially
important to allow the diameter of the coil to be smaller
than the width of the shoulders.
Wong applied conjugate gradient descent optimization to
the design of gradient coils with extended linearity.52 He
allows the position of current elements to vary to minimize
an error function. It is possible to define the error
function as desired, so it is simple to optimize over
regions of any shape, or for coil former of any shape. It
is also simple to include parameters such as coil length.
Repeated numerical evaluation of the Biot-Savart law for the
test wire positions would limit the application to coils
with a fairly small number of elements. Wong applied the
technique to the design of a local gradient coil for the
52E. C. Wong et al., Maan. Reson. Med. 21, 39, 1991.
human head.53 Its overall length was 37 cm, diameter 30 cm.
The region of interest is a cylinder 18.75 cm in diameter
and 16.5 cm long, over which the RMS (root mean square)
error in the field was less than 3% for all three axes. The
gradient coil was symmetric to avoid torque. In order to
make still shorter coils, Wong placed the return paths on a
larger cylinder.54 The wires on the inner cylinder were
connected to the return paths on the outer cylinder over
both endcaps. A coil was designed of 30 cm length, 30 cm
inner diameter, and 50 cm outer diameter. The optimization
region was a cylinder 24 cm long and 20 cm in diameter, and
the RMS error over the cylinder was 7.2%. The symmetry of
the coil eliminated the torque that arises in other short
designs. Additional points on a cylinder 70 cm in diameter
were added to the region of interest to force some partial
Another approach to the design of gradient coils that
will fit over the head is to design a coil that has its
linear region at one end. Myers and Roemer55 used only half
of a conventional coil to move the linear region to the end.
A target field approach was used by Petropoulos et al. to
design an asymmetric coil with low stored energy.56 The
coil simulated was 60 cm long and 36.4 cm in diameter. The
"center" of the coil was 14.5 cm from one end. The stored
53E. C. Wong et al., SMRM 1992, 105.
54E. C. Wong and J. S. Hyde, SMRM 1992, 583.
55C. C. Myers and P. B. Roemer, SMRM 1991, 711.
56L. S. Petropoulos et al., SMRM 1992, 4032.
energy for a gradient of 4 G/cm was calculated to be 7.93 J.
Since these coils can be made much smaller than the bore of
the magnet, eddy currents are not a serious problem and
neither of these coils is shielded. Unlike symmetric
designs, these coils experience a net torque in the magnetic
field that is potentially dangerous.
Another coil at a larger radius can be used to cancel
the torque experienced by an asymmetric gradient coil.
Petropoulos et al.57 took this approach to design a head
coil with an inner diameter of 36.4 cm, the same as their
single-layer coil described above, and an outer diameter of
48 cm. The length of both inner and outer coils was 60 cm.
The coil was designed to have a useful region that is a
sphere of 25 cm diameter. There is a price to pay in
increased stored energy, which increases over the single
layer coil value of 7.93 J to 19.2 J. Torque-compensating
windings can be added to the same cylinder as the primary
coil, resulting in a long structure one end of which is
placed over the head of the patient. Abduljalil et al.58
developed such a coil set for echo-planar imaging. The
diameter of the two radial coils was 27.2 cm and 31.2 cm.
The center of the linear region was 17 cm from the end. The
overall length was not reported, but based on artwork for
the wire pattern, it seems to be about 116 cm.
57L. S. Petropoulos et al., SMRM 1993 1305.
58A. M. Abduljalil et al., SMRM 1993, 1306.
Turner has suggested that the best approach to a
compact gradient head coil design is that of Wong, in which
the return paths are placed on a larger cylinder.59 He
points to the trapezoidal gradient coil designed by Bangert
and Mansfield,60 and discussed above, as a starting point
for this approach. The concept for such a gradient coil is
described in a patent by Frese, for a cylindrical
geometry.61 It can be thought of as a Bangert and Mansfield
coil in which the inner and outer wires have been stretched
into arcs on concentric cylinders. This is the design
independently developed by Brey and Andrew and dubbed the
Concentric Return Path Coil (CRPC). Frese suggested using a
stack of the planar CRPC units with spacing along the
cylinder's axis varied to improve size of the linear region.
He also suggested that the angle of the arcs could be varied
from plane to plane. No specific information on the spacing
or angle of the arcs is provided.
A survey of the literature suggests that it is
desirable to design a short gradient coil using the basic
concentric return path geometry to be used for the human
head. A direct error-minimization technique is appropriate
for two reasons. First, the Fourier-Bessel transform
technique, although computationally efficient, limits the
shape of the region of optimization to the surface of a
59R. Turner, Maan. Reson. Imaa. 11, 903, 1993.
60V. Bangert and P. Mansfield, J. Phys. E.: Sci. Instrum. 15, 235, 1982.
61G. Frese and E. Stetter, U. S. Patent 5,198,769, 1993.
cylinder, and axial linearity is important for the head
coil. Second, the currents are not confined to the surface
of a cylinder, and the transform technique in its present
form allows only for current on the surface of a cylinder.
An extended linear region is one of the goals of a
reduced-size gradient coil. In order to evaluate a coil
design in terms of its linear region, it is necessary to
define the boundary of the linear region. An appropriate
definition for the error associated with a field gradient
reflects the purpose of the gradient coil. In an error
minimization technique, the error definition is central to
the coil design. A reasonable parameter to use is the error
in the field, B.E.= Bz() where the desired gradient, G, is
measured at the center of the coil. Another error parameter
is the error in the gradient, G.E., defined by
G.E.= In an NMR image, error due to the
gradient coil simply produces an error in the mapping
between the sample and the image. The absolute mapping
error is simply the error in the field, B.E. In practice,
samples are usually centered in the gradient coil, so we may
want to weight the error toward the center of the coil. We
use an error parameter that corresponds to the mapping shift
relative to the component of the distance to the center in
the direction of the gradient, the relative error
B (x)- G x
R.E., defined by R.E.= Bz)- G
In order to make use of the efficiency the reduced size
of an extended-linearity gradient coil design can provide,
it is necessary to construct the coil in such a way that it
can be driven efficiently by an amplifier. By adjusting the
number of turns, it is possible to trade maximum gradient
for switching time. We will show that to obtain optimal
switching time, the amplifier should be current-controlled
to compensate for the inductance of the coil. To reduce
switching time with such an amplifier, the coil resistance
per turn should be as low as possible, even though the time
constant of the coil will be lengthened. A time-domain
model for the coil and amplifier will be used to explore the
tradeoff between maximum gradient and switching time.
Figure 14. Time-dependent voltage source v(t) drives
We show that a current-controlled amplifier gives
better switching performance into an inductive load than a
voltage source. The amplifier, modeled by a time-dependent
voltage source, v(t), is connected to a load with
resistance, Rc, and inductance, Lc, as shown in Figure 14.
When a demand is applied to a current-controlled amplifier
for some current, io, it will by definition change its
output voltage, v(t), as much and as rapidly as possible to
change the current through an inductance across the output.
If the maximum output voltage of the amplifier is Vo, and we
define the steady state output voltage v0 = Vo / Rc, where
Vo>v>O, then the amplifier output voltage and current as a
function of time will be
0 t < 0 0 t < 0
v(t) Vo 0 < t < to i(t) = I. 1- e-t/ 0 < t < to
Vo to < t < t Rc
i to < t<
where to is the time at which the output current reaches the
desired current io, and T = Lc/Rc. It is straightforward to
to = TIn Vo 
The smaller the ratio of V0 to vo, the greater the switching
time to will become. If the amplifier is a voltage source,
the desired current will never be exactly reached. It is
more desirable to use a current-controlled amplifier for
It will be shown that, with a current-controlled
amplifier, additional series resistance per turn always
decreases performance. Therefore, the series resistance
should be reduced as much as possible, for example, by using
a larger cross-sectional area for the winding in a coil of
fixed radius. Consider again the time response of a
current-controlled amplifier from Equation . Any
internal amplifier resistance can be included in Rc to avoid
any loss of generality. Assume there is some finite,
positive Rc that maximizes i(t). Then for that
R, = 0. Solving for Rc:
di V + Vo [491
di =_ i-e- L-+ -e- = 0 
dRc Rc2 c c L
and assuming that Rc and Vo do not vanish,
1 e + R/c L e = 0. 
This can be written as
(1 + x)e-x = 1 where t/T = x, 
ex = + x. 
There is no positive value of x that satisfies Equation
. Hence < 0 for all t>O, Rc>O. A lower Rc is
always an advantage when using a current-controlled
amplifier, although the time constant T = Lc/Rc of the
gradient system increases as Rc decreases. It is then
appropriate to maximize the cross-sectional area of the
windings subject to considerations of linearity and
Next we consider how the number of turns of wire in a
coil of fixed cross-sectional area can be varied to achieve
the desired performance. It is important to note that the
time constant of a coil, for a fixed area, can properly be
considered to be independent of the number of turns. This
result follows from consideration of a gradient coil at low
frequencies, as described by the equivalent circuit of a
series resistor Rc and inductor Lc as shown in Figure 15.
Figure 15. Equivalent circuit of a gradient coil in the
Let R1 be the resistance, and let L1 be the inductance of a
single turn coil. It is well known that the inductance of a
coil increases as the square of the number of turns of wire
N.62 The resistance also increases as the square of the
number of turns if the area is held constant, since as the
number of turns increases, the area of each turn diminishes.
62T. N. Trick, Introduction to Circuit Analysis, p. 256, John Wiley and
Sons, New York, 1977.
The total resistance and inductance are then
Rc = R1N2 Lc = L1N2. 
The time constant T of the coil is just the ratio
S= Lc/ R 
Perhaps surprisingly, T is independent of N. The result
does not apply when additional turns of wire are added to an
existing coil, thus increasing the area. However, since the
area should already be as large as possible to maximize the
performance, it will not be possible to increase N without
decreasing the size of the wire.
To determine how many turns of wire N should be used in
the gradient coil, we consider how rapidly and to what value
the current rises for various N, holding the area constant.
It will be shown that with a current-controlled amplifier,
the coil is optimized to switch to the field at which it
reaches a saturation current, IO, which is the maximum that
the amplifier can supply. Consider an amplifier with
negligible output impedance switching at time t = 0 from
zero current to maximum current, I0, through a gradient
coil, reaching I0 at to. Assume that Rc < VO/IO. The
current as a function of time is:
1-e1 0 t 5 to
i(t) = Rc 
TO t > to
We define a current efficiency k for a single turn so that
the gradient field G(t) = kNi(t). Rc varies as N2, and the
magnetic field G varies as N, so
f Wo1 e- 0 < t < to
G(t) = RIN 
kNIo t 2 to
A plot of G(t) for various values of N is shown in Figure
16. All three curves have the same time constant, so
0.1 0.2 0.3
Figure 16. Magnetic field produced by a current controlled
linear amplifier coupled to a coil of fixed dimensions.
Each curve represents a different number of turns.
the difference in slope is due to the relative amplitude of
the maximum gradient. The dotted line connects all the
current-limit points. Since at the current-limit point to
the amplifier is both a voltage and a current source, we can
eliminate G(to) in favor of N and to, yielding an optimum
number of turns for a given switching time.
Nopt(to) = [1[- e [57
By substituting  back into , we obtain an
expression for Gmax(to), the maximum field attainable at a
given switching time for a class of coils having the same
design except for the number of turns.
Gmax(t0) = k I 1 e-t 
Gmax(to) is just the dotted line in Figure 15. The
tradeoff between switching time and field strength is
described by the plot of Gmax(to).
In summary, a design procedure has been developed for
optimizing the switching performance of a gradient coil.
Use of a current-controlled amplifier reduces switching
time. The cross-sectional area of the winding is maximized
subject to constraints that include linearity and available
space. Then the number of turns is computed from Equation
, given the desired switching time to. The resulting
coil will give the largest possible gradient for the desired
Shielding efficiency of self-shielded gradient coils is
typically evaluated using a screening factor, a ratio of the
magnetic field outside the unshielded coil to the field
outside the shielded coil.63 It is possible to take this
63R. Turner, Maan. Reson. Imaa. 11, 903, 1993.
type of approach further and evaluate the ratio of the
gradient at the center of the coil with and without the
shield. This would seem to be a useful approach when
evaluating reduced-size gradient coils and comparing them to
shielded coils. For small eddy current fields, as in the
case of reduced-size coils, an iterative approximation
technique described below can be used to solve the integral
equation for the eddy current field. This technique is best
suited to situations where the eddy current field is much
smaller than applied field, so that a first-order
approximation can be used. However, it is simple and
To estimate the eddy current field due to a gradient
coil, we assume that there is a passive shield surrounding
the coil. The shield is typically part of the cryostat. The
boundary conditions at the shield will be
(B2 B1) n = 0 
SX (H2 H1) = K 
where B1 and H1 are the magnetic induction and field inside
the shield, B2 and H2 outside the shield, K is the surface
current on the shield, and n is an outwardly directed unit
vector normal to the surface of the shield.64 We assume the
shield is perfectly conducting, so that with H2 = 0,
H1 x n = K, 
64J. D. Jackson, Classical Electrodynamics, p. 1.5, John Wiley & Sons,
New York, 1975.
or more conveniently,
K = -B x p 
since B = o0H and n = p at the cylinder. Recall the
B(x) = J(x') x d3x'. 
4r Ix x'13
Let BO(x) be the free-space field from the gradient coil.
B(x) = B0(x) + K(x') X dx 
4 x x'13
and substituting Equation  for the surface current, for
9 = Io,
1 X X'
B(x) = Bo(x) + -- B(x') x p'] x d x 
4 |ix x'3
where p' = p(x'). This is an integral equation for B. We
can solve it iteratively. If we define Bn(x) as the field
to nth order, then the first-order solution is
B1(x) = Bo(x) + f[Bo(x') x p'] x d x. 
4x Ix x'|3
The first-order solution does not take into account
eddy currents induced by eddy currents. When the coil and
shield are in close proximity, not only are the eddy
currents larger but they are also closer to the coil, so the
second-order effect can be important. To second-order,
1 x -X d2,
B2(x) = BO(x) + [Bl(x) x '] d2x. 
4t Ix X'1
The expressions to first- and second-order for the eddy
current field will be used to evaluate numerically the eddy
current field of several coil designs. Although the result
is not exact, the expressions can easily be integrated for
coils and shields of totally arbitrary shape, assuming they
are not too close together.
The first-order calculated eddy current field of a
68.70/21.30 radial gradient coil is plotted in Figure 17.
The first-order approximation breaks down for ratios of
shield-to-coil radius of less than about 1.5.
shield radius / coil radius
1.5 2 2. 3.5 4
Figure 17. Eddy current field of 68.70/21.30 double-saddle
radial gradient coil. The field as a percentage of applied
field is plotted against the ratio of shield radius to coil
Coil projects were intended to meet experimental needs
while exploring some aspect of coil design. The 15 cm, 9 cm
and 16 mm NMR microscopy coils are well separated from any
sources of eddy currents, and demonstrate the results that
can be achieved with simple filamentary designs and without
shielding. The CRP coil development was begun to produce a
coil with good axial linearity for NMR microscopy, so that
long, narrow samples could be observed. It seemed to be
well-suited for use as a head coil for echo planar imaging,
and we turned the development toward that possible
Three Techron 7540 dual-channel amplifier units (Crown
International, Elkhorn, Illinois) are used to drive the
three-axis gradient coil sets. Each axis of the gradient
coil set is split into two halves, and one channel of each
amplifier unit is wired to each half. The plane in which
the field is always zero can be shifted slightly by varying
the relative gain in the amplifiers. This is particularly
useful in the 51 mm, 7 T magnet, since the sample is
inaccessible once loaded, and mechanical centering is
difficult. The amplifiers are rated to produce 23.8 A at 42
V direct current output. The maximum slew rate is 16 V/gs.
The output impedance is less than 7 mnQ in series with less
than 3 gH, which is negligible. The power response into a 4
0 load is +/- 1 dB up to 25 kHz for 265 W. The noise is
rated to be 112 dB below the maximum output from 20 Hz to 20
Tests of the Techron 7540 were conducted into six
loads consisting of wire-wound resistors between 1 and 9 Q.
The amplifiers were pulsed to saturation at low duty cycle.
10-90% rise times were between 4 and 6.5 9s, and so are
essentially independent of load. Thus the amplifier was
bandwidth limited, not slew-rate limited, and it is
appropriate to use a linear model. The voltage and current
60 16 T
4 U 10
o 30 U 8
20 6 4
0 I I 0
0 2 4 6 8 10 0 2 4 6 8 10
R (ohms) R (ohms)
Figure 18. Output of Techron 7540 measured into load. a)
Measured voltage; b) Calculated current.
produced are shown in Figure 18. For load resistance of
four ohms or more, the amplifier at saturation can be
modeled by a 56 V voltage source. For a load resistance of
65Crown International, Techron 7540, Elkhorn, Illinois.
four ohms or less, it can be modeled as a 15 A current
The Techron amplifiers are equipped with optional
current-control modules. With current control switched on,
an amplifier behaves like a voltage-controlled current
source. With current control switched off, an amplifier
behaves like a voltage-controlled voltage source. Current
control serves two functions when driving gradient coils.
It compensates for any variation in temperature of the
gradient coil due to resistive heating. More importantly,
it enables the coil to be switched to low fields much more
rapidly than the coil's time constant would otherwise allow.
The current-control module compares the demand (or input
signal) to the voltage across a small shunt resistor in the
output circuit. With a highly inductive load such as a
gradient coil, at high frequencies the amplifier's output
voltage is shifted almost n/2 with respect to the coil
current, and the amplifier is unstable and will oscillate.
The voltage and current response of one of the Techron
amplifiers in current mode is shown in Figure 19. The
controlled voltage overshoot reduces the current switching
time. Approximately 5 A is being switched into a 7 Q load.
An adjustable resistor-capacitor (RC) network in parallel
with the coil rolls off the high frequency gain to
compensate for the instability. The values of the RC
network are determined by the inductance of the coil. Since
the 7540 amplifiers are used with more than one coil, the
current-control units were modified so the RC networks can
be plugged in and out when gradient coils are changed.
Figure 19. Output voltage and current of Techron 7540
amplifier with current-control module. The load is the
highly inductive 9 cm field gradient coil.
The amplifier rack was equipped with wheels and shared
between the NMR microscopy and small-animal spectrometers.
It was used in voltage-control mode with the NMR microscopy
system, and current-control mode with the small animal
system where the coil inductance was much higher. Input and
output connectors were standardized to facilitate quick
conversion. A fully-shielded output cable terminated in a
fuse-and-filter chassis eliminated interference from the RF
16 mm Coil for NMR Microscopy
The 16 mm gradient coil was developed as part of the
NMR microscope development project described below. Earlier
NMR microscopy gradient coils described in the literature
were located outside of the RF probe insert, as part of the
shim coil set. A simple and straightforward approach to
improving the coil switching time, increasing the field
strength, and decreasing the eddy current field is to
integrate the gradient coils with the RF probe. This also
allows the use of a narrowbore (51 mm) magnet. Drawbacks to
this approach include a lack of flexibility. If the
gradient coil is outside the RF probe, then any RF probe can
be used. In our approach, one gradient coil is required for
each RF probe. Also, since one of the dewars associated
with the variable-temperature (VT) control system is
replaced by an acrylic tube, the range of the VT system is
reduced. Our probe did not contain any VT control
capability. The fact that the coil former was so small
encouraged us to choose a simple design to ease the
Since the sample-tube inner diameter was 4.5 mm and the
first metal tube, or shield, had an inner diameter of 33 mm,
this was a favorable case for using a reduced-size gradient
coil. A simple 68.70/21.30 radial gradient coil as
described above has a useful volume with a diameter of about
1/3 that of the coil,66 so the gradient former was chosen to
have a diameter of about 15 mm, or 5/8". A factor of two
remains in the ratio of the coil to the shield diameter.
This results in an eddy current field for the 68.70/21.30
Golay radial gradient coil, based on Figure 17, of about 20%
of the applied field.
The NMR microscope gradient coil set is of the
conventional Maxwell and Golay design described above. It
was constructed to accommodate standard 5 mm tubes used in
analytical NMR work. The 10 turns of 36 AWG enameled magnet
wire are wound on a 5/8" nominal outer diameter acrylic tube
(15.9 mm). Using a value of 1.36 Q/m for the wire67 and a
length of 0.135 m per turn, the resistance of each side of
the coil is 1.84 Q. The coil inductance can be estimated68
to be about 8 LH. The time constant of the coil is then
about 4 Js. The current efficiency of a 68.70/21.30 Golay
radial gradient coil is 0.918/a2 G/cm-A, where a is the coil
radius,69 so the coil has a current efficiency of 14.1 G/cm-
A. A Maxwell pair has a current efficiency of 0.808/a2
G/cm-A, so the coil has a current efficiency of 15.3 G/cm-A.
The typical figure for the linear region of 1/3 the diameter
of the coil is then enough to accommodate a sample. The
coil is driven by the Techron 7540 amplifier set. The coil
66F. Romeo and D. I. Hoult, Maan. Reson. Med. 1, 44, 1984.
67D. Lide,(Ed.), CRC Handbook of Chemistry and Physics, 51st Edition,
CRC Press, Boca Raton, 1970, p. 15-29.
68F. E. Terman, Radio Engineers' Handbook, McGraw-Hill, New York, 1943.
69F. Romeo and D.I. Hoult, Maan. Reson. Med. 1, 44, 1984.
has a small time constant, so using the Techron in voltage
mode does not limit the switching time. The two halves of
each gradient coil are driven separately. Since the voltage
gain of the amplifiers can be adjusted manually, it is
convenient to vary the relative gain in the coils to shift
the zero point of the magnetic field to make up for sample
The details of the coil construction are visible in
Figure 47 in the following chapter. The radial coils were
wound on a flat winding former, then removed and attached to
the acrylic tube with epoxy. To eliminate any solder
connectors within the coil, the winding former allowed two
loops to be wound at once, held apart at the correct
distance. General Electric #7031 varnish was used to hold
the wires together while the coil was being clamped to the
former. No attempt was made to arrange the wire into a
packed structure. The Maxwell pair was wound around the
radial coils. The whole assembly was potted in epoxy to
secure the coils to the former, and the 36 AWG wires were
run down to a small printed circuit board mounted to the
structure of the probe. It was necessary to pot the fine
wires to keep them from moving in the magnetic field when a
current pulse is applied.
An example of the results obtained with the coil is
reproduced in Figure 60. Although the coil is capable of
about 150 G/cm, in routine operation, the coil was operated
at a full-scale field gradient of 5 G/cm for the radial
gradients, and 10 G/cm for the axial gradient, to allow
9 cm Coil for Small Animals
An NMR magnet is frequently used for samples or animals
significantly smaller than the available bore size. It is
possible to take advantage of this fact and scale the size
of the gradient coil to match the size of the sample. One
advantage that accrues is reduced eddy current fields, since
the coil and the source of eddy currents are better
separated. Another is the increased efficiency possible
with smaller coils, since efficiency scales as the fifth
power of the diameter.70 Many applications require more
rapidly switched and more intense gradient fields than are
generally available. Diffusion-weighted imaging and
localized spectroscopy are two examples. Also, to achieve
the same bandwidth per pixel, small samples require larger
The 31 cm 2 T small-animal imaging spectrometer was
supplied with a gradient coil set manufactured by Oxford
instruments that has a clear bore of 22.5 cm, and is capable
of producing a maximum gradient of 2 G/cm with a switching
time of 1 ms. Although rat, mouse, and lizard studies, do
not require the full 22.5 cm bore, they benefit from the
horizontal orientation and will not fit into other available
magnets. Additionally, localization techniques such as
70R. Turner, Macn. Reson. Imaq. 11, 903, 1993.
selective Fourier transform71 typically require better
gradient performance than is available with a large,
unshielded gradient coil set.
To meet some of these needs, a conventional Maxwell and
68.70/21.30 Golay radial gradient coil set was designed and
constructed with a clear bore of 8.3 cm in diameter. The
useful region is a sphere of about 1/3 the diameter of the
coil, or about 3 cm. The coil was designed to accommodate
rats up to 150 g, and was able to achieve 12 G/cm with a 200
ps switching time. The intense gradients are needed for
imaging experiments on smaller samples. The coil was used
for projects involving lizards, and for development of
techniques to produce diffusion images of the spinal cord of
a rat model.
We can consider the application of the time-domain
model to the 9 cm coil. As wound, the coil will produce the
field shown in Figure 20 when driven to saturation. The
Maxwell pair, Z-axis coil, is the most efficient, followed
by the inner radial, or X-axis coil, which has better
performance than the outer radial, or Y-axis coil, because
of its smaller radius. The Maxwell pair reaches the 16 A
current limit of the amplifier, and does not increase in
field beyond that point. The radial gradient coils never
reach the current limit.
71T. H. Mareci and H. R. Brooker, J. Maan. Reson. 57, 157, 1984.
-f t (ms)
0.2 0.4 0.6 0.8 1
Figure 20. The gradient produced by the 9 cm gradient coil
set following a demand that saturates the amplifier.
Figure 21 describes the maximum field Gmax(to) possible
for switching time to for each of the three coils. The
I .. .' to (m s )
0.2 0.4 0.6 0.8 1
Figure 21. The maximum gradient that could be achieved by a
coil identical to the 9 cm coil, with the same cross-
sectional area, but with varying number of turns.
inherently lower inductance and resistance of the Maxwell
pair are reflected in its greater field. The actual and
optimal fields obtainable at 200 ps are compared in Table 1.
The Maxwell pair has about the optimal number of turns, and
its field is about the same as the optimum. The Golay coils
have about twice as many turns as optimum, and yield fields
about 80% of optimum level.
Table 1. Gradient fields for 9 cm coil set.
Gradient Actual Gradient Optimal Gradient
channel no. of after 200 no. of after 200
turns ps (G/cm) turns Rs (G/cm)
X 52 15.2 27.9 19.0
Y 52 12.3 26.9 15.9
Z 52 24.0 53.6 24.8
The radial and axial gradient coils consist of 52 turns
of AWG 27 enameled magnet wire. The wire was wound in a 7-
6-7-... close-pack configuration to minimize the cross-
sectional area of the winding, which is reduced by a factor
of 0.866 from a square winding pattern. The resulting
winding cross-section is about 2.6 mm on a side, only about
6% of the coil radius, so the winding approximates a
filament. The mean radius of the coils is 4.6 cm, 4.8 cm,
and 5.1 cm. The Maxwell pair is wound on the outside
because it is inherently more efficient and will hold down
the other coils. The two halves of each coil are wound
separately, and one channel of a stereo amplifier is wired
to each. It is driven in current mode from the Techron 7540
amplifiers. The current-control circuit helps to buck the
inductance of the coil. The coil resistance for the radial
windings is predicted to be about 6.7 Q for the inner set.
The measured resistances and time constants including the
leads and filters are given in Table 2. The time constant
of the shorted power cable with filters and fuses was too
short to measure with the amplifier, so it can be assumed to
be negligible. The last column is the inductance estimated
from the Bowtell and Mansfield formulation for coils on the
surface of cylinders.72 To adapt for the thick winding, the
height is added to the width of the winding. Calculations
for loops of square cross section compare closely to
2. Comparison of measured and predicted inductance
cm gradient coil set.
Measured Measured Time Experimental Theoretica
Resistance Constant Inductance Inductanc
(Q) (9s) (gH) (UH)
The difference between the theoretical and experimental
inductance is not due to the inductance of the power cable,
which was measured by the same technique to be about 18 pH.
It is primarily due to poor control of the cross-sectional
72R. Bowtell and P. Mansfield, Meas. Sci. Technol. 1, 431, 1990.
73F. E. Terman, Radio Engineers' Handbook, McGraw-Hill, New York, 1943.
dimensions of the winding, which expands when it is removed
from the winding former. The maximum operating field as the
system has been installed is 12.83 G/cm for X, 12.28 G/cm
for Y and 8.80 G/cm for Z. By limiting the gradient field
to a value below that corresponding to the steady state
current, the switching time is better controlled.
A drawing of the coil that illustrates how the Faraday
shield and RF coil fit together is shown in Figure 22.
RF coil Faraday Centering Maxwell Clamping
shield disk pair cam
Figure 22. Drawing of 9 cm gradient coil set with Faraday
shield and RF coil.
Figure 23 is a photograph of the assembly. The former
is an acrylic tube of 3.5" (89 mm) nominal outer diameter
and 1/8" (3.2 mm) wall thickness. The radial gradient coils
are wound on rectangular bobbins made from three flat pieces
of acrylic. The wires are held together in a hexagonal
matrix by General Electric varnish #7031 diluted in acetone.
After winding, the bobbin is disassembled and the coil is
glued onto the cylindrical former with epoxy. A variety of
Figure 23. Photograph of 9 cm gradient coil set. The power
cable and water supply cables are visible at left. The
axial and radial gradient coils are visible through the
RF coils were developed as inserts for the probe, including
19F, 1H birdcage, 31P/1H double-tuned saddle coil, and my
own 1H saddle coil. A pair of cams connected by a rod and
mounted on the edge of the mounting flanges served to lock
the probe into the magnet. A Faraday shield was used in
addition to the filter/fuse box to isolate the RF coils from
the gradient coils. The shield consisted of strips of
Reynolds heavy-duty aluminum foil approximately 2" wide,
overlapped by about 1/2", and insulated from the other
strips by masking tape that also secured the strips to
manila card stock. At one end, all strips contacted a
header strip. Provision was made to ground the shield, but
in practice it was not used. The interdigitated geometry
reduced eddy currents from the gradient coil, but allowed
the shield to serve as a barrier to the RF field. The
Maxwell pair is on the outside, which helps to hold the
radial gradient coils in place. Epoxy was initially used to
hold the windings to the former and pot the windings, but
the epoxy did not withstand the temperatures developed by
the coil and depolymerized. Polyester was selected as a
casting resin that would outperform the acrylic former under
warm conditions, and the coils were potted in polyester.
In order to cool the unit, approximately 25 m of 1/8"
O. D. by 1/64" wall polypropylene tubing was wound around
the coils. It was connected to a Neslab circulating system
that is capable of producing a pressure head of 40 psi.
Supply tubing consisted of about 30 m of 1/4" 0. D. by
1/32" wall polypropylene. Assuming laminar flow, the water
flux through a tube74 of radius r (cm) is
flux = r 4Ap
where Ap is the pressure in dyne/cm2, 1 is the length of the
tube in cm, and g is the viscosity in poise. The resulting
flux for a 40 psi drop is 8.6 cm3/s. One KW of power
transferred to the water in the tube will raise its
temperature by about 280 C. Measurement of the motion of
bubbles in the tubing reveals a flow of about 3.5 cm3/s.
74D. Lide,(Ed.), CRC Handbook of Chemistry and Physics, 51st Edition,
CRC Press, Boca Raton, 1970, p. F-34.
The difference is almost certainly due to quick-release
connectors that allow the probe to be removed or inserted at
15 cm Coil for Small Animals
The 15 cm coil was designed to accommodate larger rats
and other medium-sized laboratory animals and still produce
a higher field and faster switching time than the Oxford
gradient set. Like the 9 cm and the NMR microscope coil
set, it is based on filamentary winding design. Since it is
also driven by the Techron 7540 amplifier set, which is
under-powered for a coil of this diameter, switching
performance was at a premium. So, in contrast to the 9 cm
coil, the 15 cm coil was designed to have optimal switching
performance for the chosen switching time. In order to
provide more flexibility in choosing either high field
intensity or fast switching time, the windings were split
and could be driven either in series or parallel. Since the
coil would tend to become somewhat unwieldy as an insertable
unit, its length was the shortest that would give
essentially undiminished field intensity. Plots of the
relative error in Figures 26 and 27 illustrate the linear
region of the radial coil design. In order to avoid
aliasing signals from long animals, the axial coil was based
on an extended linearity design by Suits and Wilken.75 The
75B. H. Suits and D. E. Wilken, J. Phys. E: Sci. Instrum. 23, 565, 1989.
coil and amplifiers were capable of developing 9 G/cm in a
100 gs switching time on X, Y and Z channels.
The arcs in the 15 cm coil were arranged to have the
minimum length without losing a significant amount of
efficiency in the static limit. The standard solution of
68.70/21.30 for the arc positions arcl/arc2 leads to no
third order component from either arc, but a family of
solutions for which the third order components cancel is
available. These solutions are graphed in Figure 24.
30 40 50 60
Figure 24. The solutions to the arc position of the double-
saddle radial gradient coil.
Each solution is graphed twice, since exchanging arcl and
arc2 results in the same coil design. To improve the
relative size of the linear region to the coil, one would
like to make the coil shorter than the 68.70/21.30 solution.
The current efficiency decreases with the length, since the
return arcs tend to cancel the desired field, and moving
them closer increases the effect. However, in order to
include the effect of reducing resistive loss in the coil,
one must divide the current efficiency by the square root of
the length. The resulting measure is an indication of the
relative field that can be produced with an amplifier of a
given power. Figure 25 illustrates the relative power
efficiency as a function of the position of the return arc.
rel. power eff.
20 25 30 35 40 45 50
Figure 25. The relative power efficiency of the double-
saddle radial gradient coil as a function of the angle
between the z-axis and the current return path.
The peak efficiency is achieved at 260, but efficiency is a
weak function of angle and much shorter coils can be used
with little loss of performance. Based on this curve, the
arcs of the 15 cm radial coil were placed at 30.20 and 66.10
from the axis of the coil, compared to 21.30 and 68.70 for
the Golay coil. The overall length of the coil is reduced
by 33%. The current efficiency is 0.819/a2 G/cm-A compared
to 0.808/a2 G/cm-A for the 68.70/21.30 Golay coil. It would
be even better to use a variant of the field-versus-
switching-time approach to determine the change in
Z 0 +1%+3%+5%
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
-0.4 -0.2 0 0.2 0.4
Figure 26. Relative error plots of 30.20/66.10 radial
gradient coil. Radius of coil corresponds to 1 on scale.
a) YZ plane; b) XY plane.
Z 0 +1%+3%+5%
-1 -0.5 0 0.5 1
Figure 26--continued. Relative error plots of 30.20/66.10
radial gradient coil. Radius of coil corresponds to 1 on
scale. c) XZ plane.
performance for coils of different lengths, taking the
changing inductance into account.
The size and shape of the linear region of the 66.10
/30.20 radial gradient coil design is described by the plots
of Figures 26 and 27. It is not greatly different from the
longer 68.70/21.30 design. Note from Figure 26 that the
regions of equal relative error are not simply connected. A
three-dimensional plot of a region as large as that in the
two dimensional plots would give a false impression of the
size of the linear region, since the apparently solid
volumes would contain large holes. To avoid these "bubbles"
of linearity and give a true picture of the useful volume,
the region displayed in Figure 27 is truncated. As a result
of the truncation, it is possible to see through the linear
region. All plots were produced by direct evaluation of the
Figure 27. Perspective rendering of the 5% relative error
region of 30.20/66.10 radial gradient coil.
The axial coil was built to an extended-linearity
design proposed by Suits and Wilken.76 It consists of loops
at both 40.00 and 66.30 from the z axis. The outer loops
carry 7.5 times more current than the inner loops. Using
the additional degrees of freedom of the second-loop
position and the ratio of current in the loops, the fifth
76B. H. Suits and D. E. Wilken, J. Phys. E: Sci. Instrum. 23, 565, 1989.
and seventh order terms are canceled, resulting in about
eight times the useful volume of a Maxwell pair. For a coil
with more turns in some loops than others, the current
efficiency does not have an unambiguous definition. With
respect to the current in the outer loops, the current
efficiency of the coil is 0.635/a2 G/cm-A.
The 15 cm coil was matched to the Techron 7540
amplifiers in our laboratory using the time-domain model of
gradient performance described above. The height and width
of the windings was set to be 1 by 1 cm for the radial and 1
by 2 cm for the outer axial, to ensure that the assumption
of filamentary wires in the calculation of angle position
would be valid. The width of the outer axial winding was
increased from 1 to 2 cm, since it is farther from the
center than the others, and it was necessary to increase it
to match the radial performance. Inductance of the radial
and axial coils was calculated using a Fourier-Bessel
approach.77 The available combinations of switching time
and maximum gradient are shown in Figure 28.
In order to improve the switching time to smaller
fields, the 15 cm coil is constructed from split windings.
All the gradient coils described have the two sides driven
by separate amplifiers so that the magnetic center of the
coil can be moved. The 15 cm coil has each side split
further into two identical but electrically separate
77R. Bowtell and P. Mansfield, Meas. Sci. Technol. 1, 431, 1990.
windings. Placing the windings in parallel trades field
intensity for switching time; placing them in series reduces
switching time at the expense of lower field intensity.
0.5 1 1.5 2
Figure 28. Maximum field Gmax (G/cm) vs. switching time to
(ms) for the 15 cm field gradient coil set as driven by the
Techron 7540. Points along the curves represent designs
with different numbers of turns, increasing from left to
right. The top curve represents the inner radial coil. The
lower curves represent the outer radial coil and the axial
coil. A "slower" coil has a larger maximum field.
All three coils are optimized to approximately 10 G/cm
for the series mode. The exact number of turns for the
desired field is not used, due to the limited number of
standard wire sizes and the use of rectangular winding cross
sections. For the number of turns actually used, to and
Gmax based on the time domain model are tabulated in Table
A photograph of the 15 cm coil assembly is shown in
Figure 29. The coil is wound on an acrylic former having a
nominal O. D. of 6" (152 mm) and a wall thickness of 1/4"
(6.4 mm). The overall length was 38 mm. The radial coils
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